Integrally Closed Domains, Minimal Polynomials, and Null Ideals of Matrices

Abstract

We show that every element of the integral closure D′ of a domain D occurs as a coefficient of the minimal polynomial of a matrix with entries in D. This answers affirmatively a question of Brewer and Richman, namely, if integrally closed domains are characterized by the property that the minimal polynomial of every square matrix with entries in D is in D[x]. It follows that a domain D is integrally closed if and only if for every matrix A with entries in D the null ideal of A, N _D (A) = {f ∈ D[x] ∣ f(A) = 0} is a principal ideal of D[x].

Key Words:

• Integrally closed domain
• Minimal polynomial of a matrix
• Null ideal of a matrix
• Linear algebra over commutative rings
• Matrices over commutative rings

2000 Mathematics Subject Classification:

• Primary 13B22, 15A21
• Secondary 13B25, 12E05, 11C08, 11C20

Acknowledgments

Notes

^#Communicated by A. Prestel.